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Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 3773–3786 Research Article

On the convergence and data dependence results for multistep Picard-Mann iteration process in the class of contractive-like operators Isa Yildirima,∗, Mujahid Abbasb , Nazli Karacaa a

Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey.

b

Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan. Communicated by C. Alaca

Abstract In this paper, we introduce a new iteration process and prove the convergence of this iteration process to a fixed point of contractive-like operators. We also present a data dependence result for such mappings. c Our results unify and extend various results in the existing literature. 2016 All rights reserved. Keywords: Fixed point, contractive-like operators, convergence and data dependence. 2010 MSC: 47H10, 54H25. 1. Introduction Fixed point theory is one of the most significant tool of modern mathematics. This deals with the conditions which guarantee that a mapping T of a set X into itself admits one or more fixed points, that is, points x of X which solve an operator equation x = T x. Fixed point theory serves as an essential tool for solving problems arising in various branches of mathematical analysis. In particular, it has deep roots in nonlinear functional analysis. For instance, split feasibility problems, variational inequality problems, nonlinear optimization problems, equilibrium problems, complementarity problems, selection and matching problems, and problems of proving an existence of solution of integral and differential equations is a partial list of those problems that fall into the category of solving a fixed point problem. These problems can be modeled by the equation T x = x, where T is a nonlinear operator defined on a set equipped with some topological or order structure. ∗

Corresponding author Email addresses: [email protected] (Isa Yildirim), [email protected] (Mujahid Abbas), [email protected] (Nazli Karaca) Received 2015-01-15

I. Yildirim, M. Abbas, N. Karaca, J. Nonlinear Sci. Appl. 9 (2016), 3773–3786

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One of the basic and the most widely applied fixed point theorem in all of analysis is ”Banach (or Banach- Cassioppoli) contraction principle” due to Banach [2]. It states that if (X, d) is a complete metric space and T : X → X satisfies d(T x, T y) ≤ kd(x, y) for all x, y ∈ X with k ∈ (0, 1), then T has a unique fixed point. The basic idea of this principle rest in the use of successive approximations to establish the existence and uniqueness of solution of an operator equation T (x) = x, particularly it can be employed to prove the existence of solution of differential or integral equations. Different iterative procedures have been used to approximate the solution of fixed point problems. For instance, the sequence of Picard [17] iterates {T n x} where x is an initial guess, converges to the fixed point of a mapping T satisfying certain contractive condition. This procedure may fail to converge for some important classes of nonlinear mappings such as nonexpansive mappings, so was the justification of Mann iteration [13] procedure (see pp.8, Example 1.8 in [4]). Chidume and Mutangadura [5] showed that the Mann iterative sequence [13] fails to converge for Lipschitzian pseudocontractive mappings whereas the Ishikawa iterative procedure [9] works. It is worth mentioning that in many cases, Mann as well as Ishikawa iterative procedures converge to some fixed point of the mapping T . But, this does not hold in general. Recently, Rhoades and Soltuz ([18–20]) proved that Mann and Ishikawa iteration procedures are equivalent for several classes of mappings. It is an important subject for research to determine whether an iteration procedure converges to the fixed point of a mapping. For the results dealing with the convergence of various iteration methods, we refer to ([4, 9, 12–14, 23]). The aim of this paper is to introduce a new iteration procedure and to prove some convergence results for contractive-like operators. We also show the equivalence among convergence of iteration methods. 2. Preliminaries In the sequel the letters R and N will denote the set of all real numbers and the set of all positive integers. Let E be a nonempty closed convex subset of a Banach space X, and T : E −→ E a mapping. We denote the set of all fixed points of T by F (T ) := {p ∈ E : p = T p}. Let us first recall the following definitions and iterative procedures: Suppose that there exist real numbers a, b, c satisfying 0 < a < 1, 0 < b, c < 1/2 such that, for each pair x, y ∈ E, at least one of the following is true:   (z1 ) kT x − T yk ≤ a kx − yk , (z2 ) kT x − T yk ≤ b (kx − T xk + ky − T yk) , (2.1)  (z3 ) kT x − T yk ≤ c (kx − T yk + ky − T xk) . Such a mapping is called a Zamfirescu mapping. Zamfirescu [26] obtained an important generalization of Banach fixed point theorem using Zamfirescu mapping. It was shown in [3], the contractive condition (2.1) gives (b1 ) kT x − T yk ≤ δ kx − yk + 2δ kx − T xk if one uses (z2 ) , and (2.2) (b2 ) kT x − T yk ≤ δ kx − yk + 2δ kx − T yk if one uses (z3 ) n o b c for all x, y ∈ E, where δ := max a, 1−b , 1−c , δ ∈ [0, 1) . A mapping satisfying condition (b1 ) or (b2 ) is called a quasi-contractive mapping. This class of mappings is more general than the class of Zamfirescu mappings. Extending the above class of mappings, Osilike and Udomene [15] considered a mapping T satisfying the following contractive condition: kT x − T yk ≤ δ kx − yk + L kx − T xk

(2.3)


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for all x, y ∈ E, where L ≥ 0 and δ ∈ [0, 1) . A treatment for a similar kind of operators can be found in [1] and [11]. Further in this direction, Imoru and Olantiwo [8] gave the following definition: Definition 2.1. A self mapping T on E is called a contractive-like mapping if there exists a constant δ ∈ [0, 1) and a strictly increasing and continuous function ϕ : [0, ∞) → [0, ∞) with ϕ (0) = 0 such that the following holds kT x − T yk ≤ δ kx − yk + ϕ (kx − T xk) (2.4) for each x, y ∈ E. Remark 2.2 ([4]). It is known that condition (2.4) alone does not ensures that T has a fixed point. But if T satisfying (2.4) has a fixed point, it is certainly unique. Rhoades and S¸oltuz [20] introduced a multistep iterative algorithm as follows:  x0 ∈ E,    xn+1 = (1 − αn )xn + αn T yn1 , (2.5) yni = (1 − βni )xn + βni T yni+1 ,    r−1 yn = (1 − βnr−1 )xn + βnr−1 T xn , n ∈ N, where {αn } , {βn } , {γn } and βni , i = 1, 2, ...r − 2, r ≥ 2 are real sequences in [0, 1) satisfying certain conditions. Recently, Yildirim and Ozdemir [25] proved some convergence result by using the following multistepMann iteration process: For an arbitrary fixed order r ≥ 2,  x0 ∈ E,     xn+1 = (1 − α1n ) yn+r−2 + α1n T yn+r−2 ,     yn+r−2 = (1 − α2n ) yn+r−3 + α2n T yn+r−3 , (2.6) ..  .     yn+1 = 1 − α(r−1)n yn + α(r−1)n T yn ,    yn = (1 − αrn ) xn + αrn T xn , n ∈ N, or, in short,  x0 ∈ E,    xn+1 = (1 − α1n ) yn+r−2 + α1n T yn+r−2 , y = (1 − αin ) yn+r−(i+1) + αin T yn+r−(i+1) ,    n+r−i yn = (1 − αrn ) xn + αrn T xn , n ∈ N,

(2.7)

where {α1n } and {αin } , i = 2, ...r, are real sequence in [0, 1) . Remark 2.3. If we take r = 2 and r = 3 in (2.7), respectively, we obtain the two-step iteration procedure given in [23] and SP iteration method in [16]. S¸oltuz and Grosan [22] proved that the Ishikawa iteration [9] converges to the fixed point of T , where T : E → E is a mapping satisfying condition (2.4). In 2007, S¸oltuz [21] proved that the Mann [13], Ishikawa [9], Noor [14] and multistep (2.5) iterations are equivalent for quasi-contractive mappings in a normed space. In 2011, Chugh and Kumar [6] showed that the Picard [17], Mann [13], Ishikawa [9], new two step [23], Noor [14] and SP [16] iterations are equivalent for quasi-contractive mappings in a Banach space. In 2013, Karakaya et al. [10] proved the data dependence results for the multistep (2.5) and CR [7] iteration processes for the class of contractive-like operators satisfying (2.4). In 2013, Khan [12] introduced Picard-Mann hybrid iterative process as follows:   x0 ∈ E, xn+1 = T yn ,  yn = (1 − αn ) xn + αn T xn , n ∈ N.


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Inspired by the work of Khan [12], we introduce the following new iterative algorithm. For an arbitrary fixed order r ≥ 2,  x0 ∈ E,     xn+1 = T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] ,     yn+r−2 = T [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] , (2.8) ..  .     yn+1 = T 1 − α(r−1)n yn + α(r−1)n T yn ,    yn = T [(1 − αrn ) xn + αrn T xn ] , n ∈ N, or, in short,  x0 ∈ E,    xn+1 = T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] , y  n+r−i = T (1 − αin ) yn+r−(i+1) + αin T yn+r−(i+1) ,   yn = T [(1 − αrn ) xn + αrn T xn ] , n ∈ N,

(2.9)

P where {α1n } ⊂ [0, 1) , ∞ n=0 α1n = ∞ and {αin } ⊂ [0, 1), i = 2, ...r. Following definitions and lemmas will be needed in proving our main results. Definition 2.4 ([4]). Let T, S : X → X be two operators. We say that S is an approximate operator for T if, for some ε > 0, and any x ∈ X, the following hold: kT x − Sxk ≤ ε. Lemma 2.5 ([24]). Let {an } and {ρn } be nonnegative real sequences satisfying the following condition: an+1 ≤ (1 − µn ) an + ρn , where µn ∈ (0, 1) for all n ≥ n0 ,

P∞

n=0 µn

= ∞, and ρn = o (µn ). Then limn→∞ an = 0.

Lemma 2.6 ([22]). Let {an } be a nonnegative real sequence and there exists an n0 ∈ N such that for all n ≥ n0 satisfying the following condition: an+1 ≤ (1 − µn ) an + µn ηn , where µn ∈ (0, 1) for all n ∈ N,

P∞

n=0 µn

= ∞, and ηn ≥ 0 ∀n ∈ N. Then the following inequality holds:

0 ≤ lim sup an ≤ lim sup an ηn . n→∞

n→∞

3. Main Results Theorem 3.1. Let T : E → E be a mapping satisfying (2.4) with F (T ) 6= ∅. Let {xn } be a sequence defined n X by (2.8), where {α1n } and {αin } ⊂ [0, 1) for all i = 2, ...r satisfy the condition α1k = ∞. Then the k=0

iterative sequence {xn } converges to a unique fixed point of T . Proof. Let p ∈ F (T ). We now show that xn → p as n → ∞. From (2.4) and (2.7), we have kyn − pk = kT [(1 − αrn ) xn + αrn T xn ] − pk ≤δ k(1 − αrn ) xn + αrn T xn − pk ≤δ [(1 − αrn ) kxn − pk + αrn kT xn − pk] ≤δ [(1 − αrn ) kxn − pk + αrn δ kxn − pk + αrn ϕ (kp − T pk)] =δ [1 − αrn (1 − δ)] kxn − pk + δαrn ϕ (kp − T pk)

(3.1)


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and

kyn+1 − pk = T 1 − α(r−1)n yn + α(r−1)n T yn − p

≤δ 1 − α(r−1)n yn + α(r−1)n T yn − p ≤δ 1 − α(r−1)n kyn − pk + α(r−1)n kT yn − pk ≤δ 1 − α(r−1)n kyn − pk + α(r−1)n δ kT yn − pk + α(r−1)n ϕ (kp − T pk) =δ 1 − α(r−1)n (1 − δ) kyn − pk + δα(r−1)n ϕ (kp − T pk) .

(3.2)

Similarly, we have kyn+r−2 − pk = kT [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − pk ≤δ k(1 − α2n ) yn+r−3 + α2n T yn+r−3 − pk ≤δ [(1 − α2n ) kyn+r−3 − pk + α2n kT yn+r−3 − pk]

(3.3)

≤δ [(1 − α2n ) kyn+r−3 − pk + α2n δ kyn+r−3 − pk + α2n ϕ (kp − T pk)] =δ [1 − α2n (1 − δ)] kyn+r−3 − pk + δα2n ϕ (kp − T pk) and kxn+1 − pk = kT [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − pk ≤δ k(1 − α1n ) yn+r−2 + α1n T yn+r−2 − pk ≤δ [(1 − α1n ) kyn+r−2 − pk + α1n kT yn+r−2 − pk]

(3.4)

≤δ [(1 − α1n ) kyn+r−2 − pk + α1n δ kxn − pk + α1n ϕ (kp − T pk)] =δ [1 − α1n (1 − δ)] kyn+r−2 − pk + δα1n ϕ (kp − T pk) . By (3.1), (3.2), (3.3) and (3.4), we obtain kxn+1 − pk ≤δ [1 − α1n (1 − δ)] kyn+r−2 − pk + δα1n ϕ (kp − T pk) ≤δ [1 − α1n (1 − δ)] [δ [1 − α2n (1 − δ)] kyn+r−3 − pk + δα2n ϕ (kp − T pk)] + δα1n ϕ (kp − T pk) =δ 2 [1 − α1n (1 − δ)] [1 − α2n (1 − δ)] kyn+r−3 − pk + δ 2 [1 − α1n (1 − δ)] α2n ϕ (kp − T pk) + δα1n ϕ (kp − T pk) ≤δ 2 [1 − α1n (1 − δ)] [1 − α2n (1 − δ)] [δ [1 − α3n (1 − δ)] kyn+r−4 − pk + δα3n ϕ (kp − T pk)]

(3.5)

2

+ δ [1 − α1n (1 − δ)] α2n ϕ (kp − T pk) + δα1n ϕ (kp − T pk) .. . ≤δ r [1 − α1n (1 − δ)] [1 − α2n (1 − δ)] · · · [1 − αrn (1 − δ)] kxn − pk + δ r [1 − α1n (1 − δ)] [1 − α2n (1 − δ)] · · · 1 − α(r−1)n (1 − δ) αrn ϕ (kp − T pk) + · · · + δ 2 [1 − α1n (1 − δ)] α2n ϕ (kp − T pk) + δα1n ϕ (kp − T pk) . As ϕ (kp − T pk) = 0, so (3.5) becomes kxn+1 − pk ≤ δ r [1 − α1n (1 − δ)] [1 − α2n (1 − δ)] · · · [1 − αrn (1 − δ)] kxn − pk .

(3.6)

From {αin } ⊂ [0, 1) for i = 1, 2, ...r, we have [1 − α1n (1 − δ)] [1 − α2n (1 − δ)] · · · [1 − αrn (1 − δ)] ≤ [1 − α1n (1 − δ)] .

(3.7)


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From inequalities (3.6), (3.7) and the fact that δ ∈ (0, 1), we have kxn+1 − pk ≤ [1 − α1n (1 − δ)] kxn − pk .

(3.8)

Continuing this process in (3.8), we obtain the following inequalities: kxn − pk ≤ 1 − α1(n−1) (1 − δ) kxn−1 − pk kxn−1 − pk ≤ 1 − α1(n−2) (1 − δ) kxn−2 − pk .. . kx1 − pk ≤ [1 − α10 (1 − δ)] kx0 − pk . Therefore we have kxn+1 − pk ≤

n Y

[1 − α1k (1 − δ)] kx0 − pk .

(3.9)

k=0

Using the fact 1 − x ≤ e−x for all x ∈ [0, 1], inequality (3.9) gives kxn+1 − pk ≤

n Y

e−α1k (1−δ) kx0 − pk

k=0



n X

−1

α1k  (1−δ)   k=0 = kx0 − pkn+1  e  

(3.10) .

The result follows on taking the limit as n → ∞ on both sides of inequality (3.10). Theorem 3.2. Let T : E → E be a mapping satisfying (2.4) with F (T ) 6= ∅. If u0 = x0 ∈ E and α1n ≥ A > 0 ∀n ∈ N, then the followings are equivalent: (i) The Mann iteration [13] converges to p ∈ F (T ), (ii) The multistep Picard-Mann iteration (2.8) converges to p ∈ F (T ). Proof. We will show that (i) ⇒ (ii). Suppose that the Mann iteration [13] converges to p. Using the Mann iteration [13] and multistep Picard-Mann iteration (2.8), we have kun+1 − xn+1 k = k(1 − α1n ) un +α1n T un − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k ≤(1 − α1n ) kun −T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k + α1n kT un − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k ≤(1 − α1n ) {kun −T un k + kT un −T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k} + α1n kT un − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k =(1 − α1n ) kun −T un k + kT un − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k ≤(1 − α1n ) kun −T un k + δ kun − (1 − α1n ) yn+r−2 − α1n T yn+r−2 k + ϕ (kun −T un k) ≤(1 − α1n ) kun −T un k + δ(1 − α1n ) kun −yn+r−2 k + δα1n kun −T yn+r−2 k + ϕ (kun −T un k) ≤(1 − α1n ) kun −T un k + δ(1 − α1n ) kun −yn+r−2 k + δα1n kun −T un k + δα1n kT un −T yn+r−2 k + ϕ (kun −T un k) ≤ [1 − α1n (1 − δ)] kun −T un k + δ(1 − α1n ) kun −yn+r−2 k + δ 2 α1n kun −yn+r−2 k + δα1n ϕ (kun −T un k) + ϕ (kun −T un k) = [1 − α1n (1 − δ)] kun −T un k + δ(1 − α1n ) + δ 2 α1n kun −yn+r−2 k + (1 + δα1n )ϕ (kun −T un k) ,

(3.11)


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kun −yn+1 k = un −T 1 − α(r−1)n yn + α(r−1)n T yn

≤ kun −T un k + T un − T 1 − α(r−1)n yn + α(r−1)n T yn

≤ kun −T un k + δ un − 1 − α(r−1)n yn − α(r−1)n T yn + ϕ (kun −T un k) ≤ kun −T un k + δ 1 − α(r−1)n kun −yn k + δα(r−1)n kun −T yn k + ϕ (kun −T un k)

(3.12)

≤ kun −T un k + δ 1 − α(r−1)n kun −yn k + δα(r−1)n kun −T un k + δ 2 α(r−1)n kun −yn k + δα(r−1)n ϕ (kun −T un k) + ϕ (kun −T un k) =(1 + δα(r−1)n ) kun −T un k + δ 1 − α(r−1)n +δ 2 α(r−1)n kun −yn k + 1 + δα(r−1)n ϕ (kun −T un k) , kun −yn k ≤ kun −T [(1 − αrn ) xn + αrn T xn ]k ≤ kun −T un k + kT un − T [(1 − αrn ) xn + αrn T xn ]k ≤(1 + δαrn ) kun −T un k + δ(1 − αrn ) +δ 2 αrn kun −xn k

(3.13)

+ (1 + δαrn ) ϕ (kun −T un k) . Combining (3.11), (3.12) and (3.13), we obtain that kun+1 − xn+1 k ≤ δ(1 − α1n ) + δ 2 α1n δ(1 − α2n ) +δ 2 α2n ... δ(1 − αrn ) +δ 2 αrn kun −xn k + [1 − α1n (1 − δ)] + δ(1 − α1n ) + δ 2 α1n (1 + δα2n )+ δ(1 − α1n ) + δ 2 α1n δ(1 − α2n ) + δ 2 α2n (1 + δα3n ) + · · · + δ(1 − α1n ) + δ 2 α1n δ(1 − α2n ) + δ 2 α2n (3.14) · · · δ 1 − α(r−1)n + δ r−1 α(r−1)n (1 + δαrn ) [kun −T un k + ϕ (kun −T un k)] + α1n ϕ (kun −T un k) . Since δ ∈ (0, 1) and {αin } ⊂ [0, 1) for i = 1, 2, ...r, we have δ(1 − α1n ) + δ 2 α1n δ(1 − α2n ) +δ 2 α2n ... δ(1 − αrn ) +δ 2 αrn =δ [1 − α1n (1 − δ)] δ [1 − α2n (1 − δ)] ...δ [1 − αrn (1 − δ)]

(3.15)

≤ [1 − α1n (1 − δ)] . From inequality (3.15) and the assumption that α1n ≥ A > 0 ∀n ∈ N in (3.14), we obtain kun+1 − xn+1 k ≤ [1 − A(1 − δ)] kun −xn k + [1 − A(1 − δ)] + δ(1 − α1n ) + δ 2 α1n (1 + δα2n ) + δ(1 − α1n ) + δ 2 α1n δ(1 − α2n ) + δ 2 α2n (1 + δα3n ) + · · · + δ(1 − α1n ) + δ 2 α1n δ(1 − α2n ) + δ 2 α2n · · · δ 1 − α(r−1)n + δ r−1 α(r−1)n (1 + δαrn )} [kun −T un k + ϕ (kun −T un k)] + α1n ϕ (kun −T un k) . Set: an : = kun −xn k , µn : = A(1 − δ) ∈ (0, 1) , ρn : = [1 − A(1 − δ)] + δ(1 − α1n ) + δ 2 α1n (1 + δα2n )+ δ(1 − α1n ) + δ 2 α1n δ(1 − α2n ) + δ 2 α2n (1 + δα3n ) + · · · + δ(1 − α1n ) + δ 2 α1n δ(1 − α2n ) + δ 2 α2n · · · δ 1 − α(r−1)n + δ r−1 α(r−1)n (1 + δαrn ) [kun −T un k + ϕ (kun −T un k)] + α1n ϕ (kun −T un k) .


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Since limn→∞ kun −pk = 0 and T p = p ∈ F (T ), it follows from (2.4) that 0 ≤ kun −T un k ≤ kun −pk + kT p−T un k ≤ kun −pk + δ kp − un k + ϕ (kp−T pk) = (1 + δ) kun −pk → 0 as n → ∞. Hence limn→∞ kun −T un k = 0, that is ρn = o (µn ). It follows from Lemma 2.5 that limn→∞ kun −xn k = 0. Now kxn − pk ≤ kxn − un k + kun −pk implies that limn→∞ xn = p. Now, we will show that (ii) ⇒ (i). Suppose that multistep Picard-Mann iteration {xn } converges to p. Thus, kxn+1 − un+1 k = kxn+1 − pk + kp − un+1 k = kT p − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k + k(1 − α1n ) un +α1n T un − pk ≤δ kp − [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k + ϕ (kp − T pk) + (1 − α1n ) kun −pk + α1n kT un − T pk ≤δ(1 − α1n ) kp − yn+r−2 k + δα1n kT p − T yn+r−2 k

(3.16)

+ (1 − α1n ) kp − un k + α1n kT p − T un k ≤δ(1 − α1n ) kp − yn+r−2 k + δ 2 α1n kp − yn+r−2 k + δα1n ϕ (kp − T pk) + (1 − α1n ) kp − un k + δα1n kp − un k + α1n ϕ (kp − T pk) h i ≤ δ(1 − α1n ) + δ 2 α1n kyn+r−2 −pk + [1 − α1n (1 − δ)] kun − pk , kyn+r−2 −pk = kT [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − pk ≤ [δ k(1 − α2n ) yn+r−3 + α2n T yn+r−3 − pk + ϕ (kp − T pk)] ≤ [δ(1 − α2n ) kyn+r−3 −pk + δα2n kT yn+r−3 − pk] ≤ δ(1 − α2n ) kyn+r−3 −pk + δ 2 α2n kyn+r−3 − pk + δα2n ϕ (kp − T pk) =δ [1 − α2n (1 − δ)] kyn+r−3 −pk + [1 − α1n (1 − δ)] kun − pk

(3.17)

=δ [1 − α2n (1 − δ)] kT [(1 − α3n ) yn+r−4 + α3n T yn+r−4 ] − pk ≤··· ≤δ [1 − α2n (1 − δ)] · · · δ 1 − α(r−1)n (1 − δ) kxn − pk . Using (3.17) in (3.16), we have kxn+1 − un+1 k ≤ [1 − α1n (1 − δ)] kun − pk h i + δ(1 − α1n ) + δ 2 α1n δ [1 − α2n (1 − δ)] · · · δ 1 − α(r−1)n (1 − δ) kxn − pk . Let µn =α1n (1 − δ) an = kun − pk h i ρn = δ(1 − α1n ) + δ 2 α1n δ [1 − α2n (1 − δ)] · · · δ 1 − α(r−1)n (1 − δ) kxn − pk . Since limn→∞ kxn − pk = 0, we obtain ρn = o (µn ) . It follows from Lemma 2.5 that limn→∞ an = limn→∞ kun − pk = 0.


3781

As a sequence of Theorem 3.2, we obtain the following corollary: Corollary 3.3. Let T : E → E be a mapping satisfying (2.4) with F (T ) 6= ∅. The followings are equivalent: (i) The Picard iteration [17] converges to p ∈ F (T ), (ii) The Mann iteration [13] converges to p ∈ F (T ), (iii) The SP iteration [16] converges to p ∈ F (T ), (iv) The multistep iteration (2.5) converges to p ∈ F (T ), (v) The multistep Picard-Mann iteration (2.8) converges to p ∈ F (T ). provided that the initial guess is same for all iterations. As a consequence of Theorem 3.1, we obtain the following result which gives a relationship between the fixed point of approximate operators. Theorem 3.4. Let T : E → E be a mapping satisfying (2.4) with F (T ) 6= ∅, and S an approximate operator of T . Let {xn } and {un } be two iterative sequences defined by (2.8) associated to T and S, respectively with {α1n }, {αin } ⊂ [0, 1) for all i = 2, ...r satisfying (i) 0 ≤ αin < α1n ≤ 1 for all i = 2, ...r, and 12 ≤ α1n for all n ∈ N. n X (ii) α1k = ∞. k=0

If p = T p and q = Sq, then we have kp − qk ≤

3rε . 1−δ

Proof. For u0 ∈ E, consider the following multistep Picard-Mann iteration procedure for S:  u0 ∈ E,     un+1 = S [(1 − α1n ) vn+r−2 + α1n Svn+r−2 ] ,     v n+r−2 = S [(1 − α2n ) vn+r−3 + α2n Svn+r−3 ] , ..  .      vn+1 = S 1 − α(r−1)n vn + α(r−1)n Svn ,   vn = S [(1 − αrn ) un + αrn Sun ] , n ∈ N.

(3.18)

From (2.8), (2.4) and (3.18), we have kxn+1 − un+1 k = kT [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − S [(1 − α1n ) vn+r−2 + α1n Svn+r−2 ]k ≤ kT [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) vn+r−2 + α1n Svn+r−2 ]k + kT [(1 − α1n ) vn+r−2 + α1n Svn+r−2 ] − S [(1 − α1n ) vn+r−2 + α1n Svn+r−2 ]k ≤δ k(1 − α1n ) (yn+r−2 − vn+r−2 ) + α1n (T yn+r−2 − Svn+r−2 )k + ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) + ε ≤δ(1 − α1n ) kyn+r−2 − vn+r−2 k + δα1n kT yn+r−2 − T vn+r−2 k + δα1n kT vn+r−2 − Svn+r−2 k + ϕ(k [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] k) + ε ≤δ(1 − α1n ) kyn+r−2 − vn+r−2 k + δ 2 α1n kyn+r−2 − vn+r−2 k + δα1n ϕ (kyn+r−2 − T yn+r−2 k) + δα1n ε + ϕ(k [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] k) + ε ≤δ(1 − α1n ) kyn+r−2 − vn+r−2 k + δ 2 α1n kyn+r−2 − vn+r−2 k + δα1n ϕ (kyn+r−2 − T yn+r−2 k) + δα1n ε + ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) + ε

(3.19)


3782

=δ [1 − α1n (1 − δ)] kyn+r−2 − vn+r−2 k + δα1n ϕ (kyn+r−2 − T yn+r−2 k) + ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) + (1 + δα1n ) ε and kyn+r−2 − vn+r−2 k = kT [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − S [(1 − α2n ) vn+r−3 + α2n Svn+r−3 ]k ≤ kT [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − T [(1 − α2n ) vn+r−3 + α2n Svn+r−3 ]k + kT [(1 − α2n ) vn+r−3 + α2n Svn+r−3 ] − S [(1 − α2n ) vn+r−3 + α2n Svn+r−3 ]k ≤δ k(1 − α2n ) (yn+r−3 − vn+r−3 ) + α2n (T yn+r−3 − Svn+r−3 )k + ϕ(k [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − T [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] k) +ε ≤δ(1 − α2n ) kyn+r−3 − vn+r−3 k + δα2n kT yn+r−3 − T vn+r−3 k + δα2n kT vn+r−3 − Svn+r−3 k + ϕ (k[(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − T [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ]k) +ε ≤δ [1 − α2n (1 − δ)] kyn+r−3 − vn+r−3 k + δα2n ϕ (kyn+r−3 − T yn+r−3 k) + ϕ (k[(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − T [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ]k) + (1 + δα2n ) ε. By this induction, it follows that kyn − vn k = kT [(1 − αrn ) xn +αrn T xn ] − S [(1 − αrn ) un + αrn Sun ]k ≤ kT [(1 − αrn ) xn + αrn T xn ] − T [(1 − αrn ) un + αrn Sun ]k + kT [(1 − αrn ) un + αrSun ] − S [(1 − αrn ) un + αrn Sun ]k ≤δ k(1 − αrn ) (xn −un ) + αrn (T xn −Sun )k + ϕ (k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn ) xn + αrn T xn ]k) +ε ≤δ(1 − αrn ) kxn −un k + δαrn kT xn −T un k + δαrn kT un −Sun k + ϕ (k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn )xn +αrn T xn ]k) +ε ≤δ [1 − αrn (1 − δ)] kxn −un k + δαrn ϕ (kxn −T xn k) + ϕ (k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn )xn +αrn T xn ]k) + (1 + δαrn ) ε. Thus, using this inequality in (3.19), we get kxn+1 − un+1 k ≤δ [1 − α1n (1 − δ)] {δ [1 − α2n (1 − δ)] kyn+r−3 − vn+r−3 k + δα2n ϕ (kyn+r−3 − T yn+r−3 k) + ϕ (k[(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − T [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ]k) + (1 + δα2n ) ε} + δα1n ϕ (kyn+r−2 − T yn+r−2 k) + ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) + (1 + δα1n ) ε =δ [1 − α1n (1 − δ)] δ [1 − α2n (1 − δ)] kyn+r−3 − vn+r−3 k


3783

+ δα1n ϕ (kyn+r−2 − T yn+r−2 k) + δ [1 − α1n (1 − δ)] δα2n ϕ (kyn+r−3 − T yn+r−3 k) + ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) + (1 + δα1n ) ε + δ [1 − α1n (1 − δ)] (1 + δα2n ) ε ≤··· ≤δ [1 − α1n (1 − δ)] δ [1 − α2n (1 − δ)] · · · δ [1 − αrn (1 − δ)] kxn − un k + δα1n ϕ (kyn+r−2 − T yn+r−2 k) + δ [1 − α1n (1 − δ)] δα2n ϕ (kyn+r−3 − T yn+r−3 k) + · · · + δ [1 − α1n (1 − δ)] · · · δ 1 − α(r−1)n (1 − δ) δαrn ϕ (kxn −T xn k)

(3.20)

+ ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) + δ [1 − α1n (1 − δ)] ϕ (k[(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − T [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ]k) + · · · + δ [1 − α1n (1 − δ)] · · · δ 1 − α(r−1)n (1 − δ) ϕ (k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn )xn +αrn T xn ]k) + (1 + δα1n ) ε + δ [1 − α1n (1 − δ)] (1 + δα2n ) ε + · · · + δ [1 − α1n (1 − δ)] · · · δ 1 − α(r−1)n (1 − δ) (1 + δαrn ) ε. Since δ ∈ [0, 1) and {α1n }, {αin } ⊂ [0, 1) for all i = 2, ...r, we have δ [1 − α1n (1 − δ)] δ [1 − α2n (1 − δ)] · · · δ [1 − αrn (1 − δ)] ≤ [1 − α1n (1 − δ)] , 1 − α1n ≤ α1n , δ [1 − α1n (1 − δ)] < 1. Using the above inequality and the assumption (i) in (3.20), we get kxn+1 − un+1 k ≤ [1 − α1n (1 − δ)] kxn − un k + α1n ϕ (kyn+r−2 − T yn+r−2 k) + α1n ϕ (kyn+r−3 − T yn+r−3 k) + · · · + α1n ϕ (kxn −T xn k) + 2α1n ϕ(k [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] k) + 2α1n ϕ(k [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − T [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] k) + · · · + 2α1n ϕ (k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn )xn +αrn T xn ]k) + r [(1 + δα1n ) ε] ≤ [1 − α1n (1 − δ)] kxn − un k ϕ (kyn+r−2 − T yn+r−2 k) + · · · + ϕ (kxn −T xn k) α1n (1 − δ) { (1 − δ) 2ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) (1 − δ) 2ϕ (k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn )xn +αrn T xn ]k) + ··· + (1 − δ) 3rε + }. (1 − δ)

(3.21)


3784

Let an = kxn − un k , µn =α1n (1 − δ) , ϕ (kyn+r−2 − T yn+r−2 k) + · · · + ϕ (kxn −T xn k) ηn = (1 − δ) 2ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) + (1 − δ) 2ϕ (k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn )xn +αrn T xn ]k) + ··· + (1 − δ) 3rε + . (1 − δ) From Theorem 3.1 it follows that limn→∞ kxn − pk = 0. Since T satisfies condition (2.4) and T p = p ∈ F (T ). Therefore 0 ≤ kxn − T xn k ≤ kxn − pk + kT p − T xn k ≤ kxn − pk + δ kp − xn k + ϕ (kp − T pk) =(1 + δ) kxn − pk → 0 as n → ∞. By (2.4) and (2.8) and the fact that α1n , αin ∈ [0, 1) ∀n ∈ N, i = 2, ..., r, we have 0 ≤ kyn+r−2 − T yn+r−2 k ≤ kyn+r−2 − pk + kT p − T yn+r−2 k ≤ kyn+r−2 − pk + δ kp − yn+r−2 k + ϕ (kp − T pk) =(1 + δ) kyn+r−2 − pk =(1 + δ) kT [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] −pk ≤(1 + δ)δ k(1 − α2n ) yn+r−3 + α2n T yn+r−3 − pk ≤(1 + δ)δ [(1 − α2n ) kyn+r−3 − pk + α2n kT yn+r−3 − pk] ≤(1 + δ)δ [(1 − α2n ) kyn+r−3 − pk + α2n δ kyn+r−3 − pk + α2n ϕ (kp − T pk)] =(1 + δ)δ [1 − α2n (1 − δ)] kyn+r−3 − pk =(1 + δ)δ [1 − α2n (1 − δ)] kT [(1 − α3n ) yn+r−4 + α3n T yn+r−4 ] − pk ≤··· ≤(1 + δ)δ [1 − α2n (1 − δ)] · · · δ 1 − α(r−1)n (1 − δ) kxn − pk → 0 as n → ∞. Above also holds for kyn+r−3 − T yn+r−3 k , · · · , kyn − T yn k . As ϕ is continuous, so we obtain lim ϕ (kxn −T xn k) = lim ϕ (kyn+r−2 − T yn+r−2 k) = · · · = lim ϕ (kyn − T yn k) = 0.

n→∞

n→∞

n→∞

From (2.4) and (3.2), we have 0 ≤ k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k ≤ k(1 − α1n ) (yn+r−2 − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ])k + kα1n (T yn+r−2 − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ])k ≤(1 − α1n ) kyn+r−2 − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k

I. Yildirim, M. Abbas, N. Karaca, J. Nonlinear Sci. Appl. 9 (2016), 3773–3786 + α1n δ kyn+r−2 − [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k + α1n ϕ (kyn+r−2 − T yn+r−2 k) =(1 − α1n ) kyn+r−2 − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k 2 + α1n δ kyn+r−2 − T yn+r−2 k + α1n ϕ (kyn+r−2 − T yn+r−2 k)

≤(1 − α1n ) [kyn+r−2 − pk + kT p − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k] 2 + α1n δ kyn+r−2 − T yn+r−2 k + α1n ϕ (kyn+r−2 − T yn+r−2 k)

≤(1 − α1n ) [kyn+r−2 − pk + δ kp − [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k + ϕ (kp − T pk)] 2 + α1n δ kyn+r−2 − T yn+r−2 k + α1n ϕ (kyn+r−2 − T yn+r−2 k)

≤(1 − α1n ) [kyn+r−2 − pk + δ(1 − α1n ) kyn+r−2 − pk + δα1n kT p − T yn+r−2 k] 2 + α1n δ kyn+r−2 − T yn+r−2 k + α1n ϕ (kyn+r−2 − T yn+r−2 k) ≤(1 − α1n ) (1 + δ(1 − α1n )) kyn+r−2 − pk + δ 2 α1n kp − yn+r−2 k + δα1n ϕ (kp − T pk) 2 + α1n δ kyn+r−2 − T yn+r−2 k + α1n ϕ (kyn+r−2 − T yn+r−2 k) =(1 − α1n ) δ 2 α1n −δα1n + δ + 1 kyn+r−2 − pk 2 + α1n δ kyn+r−2 − T yn+r−2 k + α1n ϕ (kyn+r−2 − T yn+r−2 k) .

By this induction, 0 ≤ k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn )xn +αrn T xn ]k ≤(1 − αrn ) kxn − T [(1 − αrn )xn +αrn T xn ]k 2 + αrn δ kxn − T xn k + αrn ϕ (kxn − T xn k)

≤(1 − αrn ) [kxn − pk + kT p − T [(1 − αrn )xn +αrn T xn ]k] 2 + αrn δ kxn − T xn k + αrn ϕ (kxn − T xn k)

≤(1 − αrn ) [kxn − pk + δ kp − [(1 − αrn )xn +αrn T xn ]k + ϕ (kp − T pk)] 2 + αrn δ kxn − T xn k + αrn ϕ (kxn − T xn k)

≤(1 − αrn ) [kxn − pk + δ(1 − αrn ) kxn − pk + δαrn kT p − T xn k] 2 + αrn δ kxn − T xn k + αrn ϕ (kxn − T xn k) ≤(1 − αrn ) (1 + δ(1 − αrn )) kxn − pk + δ 2 αrn kp − xn k + δαrn ϕ (kp − T pk) 2 + αrn δ kxn − T xn k + αrn ϕ (kxn − T xn k) =(1 − αrn ) δ 2 αrn − δαrn + δ + 1 kxn − pk 2 + αrn δ kxn − T xn k + αrn ϕ (kxn − T xn k) .

Since lim kyn+r−2 − pk = lim kyn+r−3 − pk = · · · = lim kxn − pk = 0

n→∞

n→∞

n→∞

and lim kyn+r−2 − T yn+r−2 k = lim kyn+r−3 − T yn+r−3 k = · · · = lim kxn − T xn k = 0,

n→∞

n→∞

n→∞

we get lim ϕ (k[(1 − α1n ) yn+r−2 + α1n T yn+r−2 ] − T [(1 − α1n ) yn+r−2 + α1n T yn+r−2 ]k) = 0,

n→∞

lim ϕ (k[(1 − α2n ) yn+r−3 + α2n T yn+r−3 ] − T [(1 − α2n ) yn+r−3 + α2n T yn+r−3 ]k) = 0,

n→∞

.. . lim ϕ (k[(1 − αrn )xn +αrn T xn ] − T [(1 − αrn )xn +αrn T xn ]k) = 0.

n→∞

3785


3786

Hence, from Lemma 2.6, we obtain that 0 ≤ lim sup kxn − un k ≤ lim sup ηn = n→∞

n→∞

3rε . 1−δ

Applying Lemma 2.6 to (3.21) implies that kp − qk ≤

3rε . 1−δ

References [1] M. Abbas, P. Vetro, S. H. Khan, On fixed points of Berinde’s contractive mappings in cone metric spaces, Carpathian J. Math., 26 (2010), 121–133. 2 [2] S. Banach, Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math., 3 (1922), 133–181. 1 [3] V. Berinde, On the convergence of the Ishikawa iteration in the class of quasi contractive operators, Acta Math. Univ. Comenianae, 73 (2004), 119–126. 2 [4] V. Berinde, Iterative approximation of fixed points, Springer Berlin Heidelberg, New York, (2007). 1, 2.2, 2.4 [5] C. E. Chidume, S. A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc., 129 (2001), 2359–2363. 1 [6] R. Chugh, V. Kumar, Strong convergence of SP iterative scheme for quasi-contractive operators in Banach spaces, Int. J. Comput. Appl., 31 (2011), 21–27. 2 [7] R. Chugh, V. Kumar, S. Kumar, Stronge converge of a new three step iterative scheme in Banach spaces, Amer. J. Comput. Math., 2 (2012), 345–357. 2 [8] C. O. Imoru, M. O. Olantiwo, On the stability of Picard and Mann iteration processes, Carpathian J. Math., 19 (2003), 155–160. 2 [9] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. 1, 2 [10] V. Karakaya, F. G¨ ursoy, K. Do˘ gan, M. Ert¨ urk, Data dependence results for multistep and CR iterative schemes in the class of contractive-like operators, Abstr. Appl. Anal., 2013 (2013), 7 pages. 2 [11] S. H. Khan, Common fixed points of quasi-contractive type operators by a generalized iterative process, IAENG Int. J. Appl. Math., 41 (2011), 260–264. 2 [12] S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl., 2013 (2013), 10 pages. 1, 2 [13] W. R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc., 4 (1953), 506–510. 1, 2, 3.2, 3, 3.3 [14] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. 1, 2 [15] M. O. Osilike, A. Udomene, Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings, Indian J. Pure Appl. Math., 30 (1999), 1229–1234. 2 [16] W. Phuengrattana, S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235 (2011), 3006–3014. 2.3, 2, 3.3 [17] E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl., 6 (1890), 145–210. 1, 2, 3.3 [18] B. E. Rhoades, S. M. S ¸ oltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map, J. Math. Anal. Appl., 283 (2003), 681–688. 1 [19] B. E. Rhoades, S. M. S ¸ oltuz, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Int. J. Math. Math. Sci., 42 (2003), 2645–2651. [20] B. E. Rhoades, S. M. S ¸ oltuz, The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonlinear Anal., 58 (2004), 219–228. 1, 2 [21] S. M. S ¸ oltuz, The equivalence between Krasnoselskij, Mann, Ishikawa, Noor and multistep iterations, Math. Commun., 12 (2007), 53–61. 2 [22] S. M. S ¸ oltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl., 2008 (2008), 7 pages. 2, 2.6 [23] S. Thianwan, Common fixed points of new iterations for two asymptotically nonexpansive nonself mappings in a Banach space, J. Comput. Appl. Math., 224 (2009), 688–695. 1, 2.3, 2 [24] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc., 113 (1991), 727–731. 2.5 [25] I. Yildirim, M. Ozdemir, A new iterative process for common fixed points of finite families of non-selfasymptotically non-expansive mappings, Nonlinear Anal., 71 (2009), 991–999. 2 [26] T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math., 23 (1972), 292–298. 2